Module 8 Vocab

Question 1

PIE for 3 Events

Answer 1

Pr[A∪B∪C] = Pr[A] + Pr[B] + Pr[C] - Pr[A∩B] - Pr[B∩C] - Pr[C∩A] - Pr[A∩B∩C]

Question 2

Independent

Answer 2

Two events A,B ⊆ Ω
A⊥B when Pr[A∩B] = Pr[A]⋅Pr[B]

Question 3

Independence is ___________

Answer 3

A⊥B iff B⊥A

Question 4

Another way to think of A∩B is…

Answer 4

both A and B are happening

Question 5

Property Ind i

Answer 5

If Pr[A] = 0, then A⊥B for any B

Question 6

Property Ind ii

Answer 6

Ω⊥E for any E

Question 7

Alternate Property Ind i

Answer 7

∅⊥E for any E

Question 8

Property Ind iii

Answer 8

If A⊥B, then Pr[A∪B] = 1 - (1 - Pr[A])(1 - Pr[B])

Question 9

Property Ind iv

Answer 9

A⊥B iff ˉA⊥B iff A⊥ˉB iff ˉA⊥ˉB

Question 10

Property Ind iv (words)

Answer 10

A and B are independent iff the complement of each event is independent of the other event and their complements are independent

Question 11

Independent vs. Disjoint

Answer 11

disjoint events are typically not independent of each other

Question 12

If A,B are independent and disjoint, then…

Answer 12

at least one of A,B has a probability of 0

Question 13

If E⊥ˉE

Answer 13

then Pr[E] = 0 or 1

Question 14

Any event of probability 0,1 has the property

Answer 14

X⊥X

Question 15

If Pr[A] = 0, then A⊥B for any B

Answer 15

Property Ind i

Question 16

A⊥B iff ˉA⊥B iff A⊥ˉB iff ˉA⊥ˉB

Answer 16

Property Ind iv

Question 17

If A⊥B, then Pr[A∪B] = 1 - (1 - Pr[A])(1 - Pr[B])

Answer 17

Property Ind iii

Question 18

Ω⊥E for any E

Answer 18

Property Ind ii

Question 19

∅⊥E for any E

Answer 19

Alternate Property Ind i

Question 20

Two Independent Bernoulli Trials

Answer 20

SS would be p^2, SF and FS would be pq and FF would be q^2

Question 21

“at least one success and at least one failure”

Answer 21

count complementarily where the “bad” cases are all successes and all failures

Question 22

NOT Independent 1

Answer 22

A⊥B, B⊥C, C⊥A but Pr[A∩B∩C] ≠ Pr[A]⋅Pr[B]⋅Pr[C]

Question 23

NOT Independent 2

Answer 23

Pr[A∩B∩C] = Pr[A]⋅Pr[B]⋅Pr[C] but A∤B, B∤C, C∤A
(∤ means not independent)

Question 24

Pairwise Independent

Answer 24

Events A1,…,An are PI when for any distinct i and j between 1 and n we have Ai⊥Aj

Question 25

Mutually Independent

Answer 25

Events A1,…An are MI when for any {i1,…,ik} ⊆ [1..n]
we have Pr[Ai1,∩…∩Aik] = Pr[Ai1]⋅…⋅Pr[Aik]

Question 26

Pairwise v. Mutual Independence

Answer 26

MI implies PI
so if something is not PI then it can’t be MI either

Question 27

Generalized Ind iii

Answer 27

for unions of mutually independent events
Pr[A1∪…∪An] = 1 - (1 - Pr[A1]) ⋅…⋅ (1 - Pr[An])

Question 28

Set intersection distributes over set union

Answer 28

A∩(B∪C) = (A∩B)∪(A∩C)

Question 29

Complements are ________

Answer 29

DISJOINT

Question 30

You can see ∩ as

Answer 30

and
like A∩B means events A and B are happening

Question 31

DeMorgan’s Lemma

Answer 31

ˉˉˉA∪B∪C = ˉA∩ˉB∩ˉC

Question 32

PIE and Unions Connection

Answer 32

Rolling 10 fair dice independently and rolling a fair die 10 times independently both give rise to the same probability space

Question 33

When to use conditional probability

Answer 33

Given (Ω, Pr) we are interested in event E in a context in which we already know for sure that event U happened/is happening

Question 34

E | U

Answer 34

E conditioned on U

Question 35

Pr[E|U]

Answer 35

Pr[E∩U] / Pr[U] provided Pr[U] ≠ 0

Question 36

Chain Rule (for 3 events)

Answer 36

For any events A,B,C in the same space we have:
Pr[A∩B∩C] = Pr[A] ⋅ Pr[B|A] ⋅ Pr[C|A∩B]

Question 37

For any two events A,B in the same probability space, the following two statements are equivalent

Answer 37

(i) A⊥B
(ii) Pr[B] = 0 or (Pr[B] ≠ 0 and Pr[A|B] = Pr[A])

Question 38

Chain Rule (for 2 events)

Answer 38

Pr[A∩B] = Pr[A] ⋅ Pr[B|A]

Question 39

_________ is a particular case of the formula for ________________ probability

Answer 39

Independence, conditional

Question 40

Generalized Chain Rule

Answer 40

For any events A1,…,An in the same space we have
Pr[A1∩…∩An] = Pr[A1] ⋅ Pr[A2|A1] ⋅ Pr[A3|A1∩A2] ⋅ … ⋅ Pr[An|A1∩…∩An-1]

Question 41

A ⊆ B

Answer 41

A∩B = A

Question 42

What does disjoint mean about A∩B?

Answer 42

A∩B = ∅

Question 43

Pr[A∩B] =

Answer 43

Pr[A|B] ⋅ Pr[B]

Question 44

In general, the branches are labeled with __________ probabilities and along each branch the _______ _____ computes the probability of the ______________.

Answer 44

conditional
chain rule
outcome